36,099 research outputs found
A Bayesian Approach to Sparse Model Selection in Statistical Shape Models
Groupwise registration of point sets is the fundamental step in creating statistical shape models (SSMs). When the number of points on the sets varies across the population, each point set is often regarded as a spatially transformed Gaussian mixture model (GMM) sample, and the registration problem is formulated as the estimation of the underlying GMM from the training samples. Thus, each Gaussian in the mixture specifies a landmark (or model point), which is probabilistically corresponded to a training point. The Gaussian components, transformations, and probabilistic matches are often computed by an expectation-maximization (EM) algorithm. To avoid over- and under-fitting errors, the SSM should be optimized by tuning the required number of components. In this paper, rather than manually setting the number of components before training, we start from a maximal model and prune out the negligible points during the registration by a sparsity criterion. We show that by searching over the continuous space for optimal sparsity level, we can reduce the fitting errors (generalization and specificities), and thereby help the search process for a discrete number of model points. We propose an EM framework, adopting a symmetric Dirichlet distribution as a prior, to enforce sparsity on the mixture weights of Gaussians. The negligible model points are pruned by a quadratic programming technique during EM iterations. The proposed EM framework also iteratively updates the estimates of the rigid registration parameters of the point sets to the mean model. Next, we apply the principal component analysis to the registered and equal-length training point sets and construct the SSMs. This method is evaluated by learning of sparse SSMs from 15 manually segmented caudate nuclei, 24 hippocampal, and 20 prostate data sets. The generalization, specificity, and compactness of the proposed model favorably compare to a traditional EM based model
Spike-and-Slab Priors for Function Selection in Structured Additive Regression Models
Structured additive regression provides a general framework for complex
Gaussian and non-Gaussian regression models, with predictors comprising
arbitrary combinations of nonlinear functions and surfaces, spatial effects,
varying coefficients, random effects and further regression terms. The large
flexibility of structured additive regression makes function selection a
challenging and important task, aiming at (1) selecting the relevant
covariates, (2) choosing an appropriate and parsimonious representation of the
impact of covariates on the predictor and (3) determining the required
interactions. We propose a spike-and-slab prior structure for function
selection that allows to include or exclude single coefficients as well as
blocks of coefficients representing specific model terms. A novel
multiplicative parameter expansion is required to obtain good mixing and
convergence properties in a Markov chain Monte Carlo simulation approach and is
shown to induce desirable shrinkage properties. In simulation studies and with
(real) benchmark classification data, we investigate sensitivity to
hyperparameter settings and compare performance to competitors. The flexibility
and applicability of our approach are demonstrated in an additive piecewise
exponential model with time-varying effects for right-censored survival times
of intensive care patients with sepsis. Geoadditive and additive mixed logit
model applications are discussed in an extensive appendix
A Noise-Robust Fast Sparse Bayesian Learning Model
This paper utilizes the hierarchical model structure from the Bayesian Lasso
in the Sparse Bayesian Learning process to develop a new type of probabilistic
supervised learning approach. The hierarchical model structure in this Bayesian
framework is designed such that the priors do not only penalize the unnecessary
complexity of the model but will also be conditioned on the variance of the
random noise in the data. The hyperparameters in the model are estimated by the
Fast Marginal Likelihood Maximization algorithm which can achieve sparsity, low
computational cost and faster learning process. We compare our methodology with
two other popular learning models; the Relevance Vector Machine and the
Bayesian Lasso. We test our model on examples involving both simulated and
empirical data, and the results show that this approach has several performance
advantages, such as being fast, sparse and also robust to the variance in
random noise. In addition, our method can give out a more stable estimation of
variance of random error, compared with the other methods in the study.Comment: 15 page
Exact Dimensionality Selection for Bayesian PCA
We present a Bayesian model selection approach to estimate the intrinsic
dimensionality of a high-dimensional dataset. To this end, we introduce a novel
formulation of the probabilisitic principal component analysis model based on a
normal-gamma prior distribution. In this context, we exhibit a closed-form
expression of the marginal likelihood which allows to infer an optimal number
of components. We also propose a heuristic based on the expected shape of the
marginal likelihood curve in order to choose the hyperparameters. In
non-asymptotic frameworks, we show on simulated data that this exact
dimensionality selection approach is competitive with both Bayesian and
frequentist state-of-the-art methods
Normal-Mixture-of-Inverse-Gamma Priors for Bayesian Regularization and Model Selection in Structured Additive Regression Models
In regression models with many potential predictors, choosing an appropriate subset of covariates and their interactions at the same time as determining whether linear or more flexible functional forms are required is a challenging and important task. We propose a spike-and-slab prior structure in order to include or exclude single coefficients as well as blocks of coefficients associated
with factor variables, random effects or basis expansions
of smooth functions. Structured additive models with this prior structure are estimated with Markov Chain Monte Carlo using a redundant multiplicative parameter expansion. We discuss shrinkage properties of the novel prior induced by the redundant parameterization, investigate its sensitivity to hyperparameter settings and compare performance of the proposed method in terms of model selection, sparsity recovery, and estimation error for Gaussian, binomial and Poisson responses on real and simulated data sets with that of component-wise boosting and other approaches
Sparse Estimation using Bayesian Hierarchical Prior Modeling for Real and Complex Linear Models
In sparse Bayesian learning (SBL), Gaussian scale mixtures (GSMs) have been
used to model sparsity-inducing priors that realize a class of concave penalty
functions for the regression task in real-valued signal models. Motivated by
the relative scarcity of formal tools for SBL in complex-valued models, this
paper proposes a GSM model - the Bessel K model - that induces concave penalty
functions for the estimation of complex sparse signals. The properties of the
Bessel K model are analyzed when it is applied to Type I and Type II
estimation. This analysis reveals that, by tuning the parameters of the mixing
pdf different penalty functions are invoked depending on the estimation type
used, the value of the noise variance, and whether real or complex signals are
estimated. Using the Bessel K model, we derive a sparse estimator based on a
modification of the expectation-maximization algorithm formulated for Type II
estimation. The estimator includes as a special instance the algorithms
proposed by Tipping and Faul [1] and by Babacan et al. [2]. Numerical results
show the superiority of the proposed estimator over these state-of-the-art
estimators in terms of convergence speed, sparseness, reconstruction error, and
robustness in low and medium signal-to-noise ratio regimes.Comment: The paper provides a new comprehensive analysis of the theoretical
foundations of the proposed estimators. Minor modification of the titl
Bayesian nonparametric sparse VAR models
High dimensional vector autoregressive (VAR) models require a large number of
parameters to be estimated and may suffer of inferential problems. We propose a
new Bayesian nonparametric (BNP) Lasso prior (BNP-Lasso) for high-dimensional
VAR models that can improve estimation efficiency and prediction accuracy. Our
hierarchical prior overcomes overparametrization and overfitting issues by
clustering the VAR coefficients into groups and by shrinking the coefficients
of each group toward a common location. Clustering and shrinking effects
induced by the BNP-Lasso prior are well suited for the extraction of causal
networks from time series, since they account for some stylized facts in
real-world networks, which are sparsity, communities structures and
heterogeneity in the edges intensity. In order to fully capture the richness of
the data and to achieve a better understanding of financial and macroeconomic
risk, it is therefore crucial that the model used to extract network accounts
for these stylized facts.Comment: Forthcoming in "Journal of Econometrics" ---- Revised Version of the
paper "Bayesian nonparametric Seemingly Unrelated Regression Models" ----
Supplementary Material available on reques
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